Accuracy, uncertainty and linear alteration of geographic data (2)

In the previous article we discussed the different geographic coordinate systems: astronomical, geocentric and geodetic coordinates. Let’s have a look at the possible sources of error.

Ellipsoidal and real distance

Now, let’s see a practical case. You invest in a GPS that allows an accuracy of 10 cm. You map a network that is one kilometre long. Question: What is the difference between the actual positions of the first and the last point of the network?

There are two answers depending on whether you are using geocentric or geodesic coordinates. In the first case, the answer is: maximum 10 cm. In the second, the answer is: it depends on the place, but, obviously, it is more than 10 cm and you can be surprised to find yourself with tens of meters!

Let’s see the explanation:

The starting point of the network is M and the arrival point is M ‘, you see in the figure that the real distance between the points is M-M’, which you can calculate exactly with geocentric coordinates, and if you use the geodesic coordinates you can calculate the distance P-P ‘, clearly different from the real distance. The more distant the points are, the greater the difference.

Ellipsoidal precision

Geodetic systems were designed centuries before the advent of GPS and electronic measurements. Since the modelling of the ellipsoid is a mathematical process, it has an intrinsic precision that cannot be exceeded. This precision, for geodetic systems which were used throughout the world until the 90s, is a metric precision. What does that mean? It means that a longitude latitude expressed on an IGN map in Lambert 2 NTF, where a position on a chart in Mercator Europe 50, or a UTM WGS84, has an uncertainty of one meter due, only, to the fact that this calculation locates this place on the ‘ellipsoid’.

It is because of this precision, largely sufficient with the old positioning means, but unsuitable for satellite positioning, that a new, more accurate geodetic system has been put in place.

This is the reason for the withdrawal in France of the NTF system and the adoption of the RGF93.

New global geodetic systems

There are two types of geodetic systems:

The terrestrial systems, established before the advent of space, also called local systems because they are set up for a specific region or country. These systems are the least accurate and their achievements are no longer updated. They are, now, replaced by

the space systems also called global systems because they rely on space techniques that cover the entire planet.

A reference ellipsoid is used to define the Terrestrial systems, a primary astronomically observed point and an original meridian. Therefore, in these systems, a point on the Earth’s surface is identified by two-dimensional coordinates (longitude and latitude). The centre of the system may deviate several hundred meters from the centre of the land mass.
Space systems are determined from astronomical and geodetic fundamental constants. In these systems, a point on the Earth’s surface is identified by three-dimensional coordinates (longitude, latitude, and ellipsoidal height).
The IAG GRS 1980 ellipsoid (International Association of Geodesy, Geodetic Reference System 1980) is the international ellipsoid defined by the International Earth Rotation and the Reference System Service (IERS). It is almost identical to the ellipsoid WGS84 (World Geodetic System 1984) because it differs only a tenth of a millimetre on the half minor axis.

RGF93 is the French part of the European Terrestrial Reference System (ETRS89) for the period 1989,0, the European section of the ITRS89 (International Terrestrial Reference System).
It should be noted that these three systems are completely coherent and offer an accuracy for the coordinates of the order of 2 cm.
Thus, in GIS software, these three systems can be used without risk of any problem as to the quality of the georeferencing.
The American WGS84 system is the GPS reference system. It was developed from Doppler measurements metric precision.
This system is consistent with the RGF93, ETRS as well as ITRS systems, but with metric accuracy. Therefore, when converting data from the old French NTF system to RGF93 or WGS84, the coordinates may differ by a few decimals.

Projection and linear alteration

The most accurate position available is that of a good GPS. But this measurement is calculated in angles (latitude / longitude) and positions our points on a three-dimensional surface (the ellipsoid).
The use of maps and planar representations of our planet are the most widespread. We have known for a long time the multiple reasons: a map is easier to handle than a globe, the metric coordinates become more easily exploitable than angular values ​​often expressed in the sexagesimal system, and measuring a distance is easier on a flat surface, even if the measurement is tainted by error.

Therefore, we resort to a mathematical projection to represent all or part of an ellipsoidal model of the terrestrial surface, on a plane.
Since it is impossible to rigorously develop a portion of sphere or ellipsoid on a plane (flattening of the skin of an orange on a plane), all projections introduce deformations that alter all or part of the elements of the area to be represented: lengths, angles or surfaces. Nevertheless, it is possible, by carefully defining the parameters of the projection, to minimize certain deformations. Three types of projections are defined:
conforming projections that hold angles, equivalent projections that hold surfaces, or aphylactic projections which are compromise solutions that best compensate for various alterations.

The Lambert and Mercator projections used in mainland France and in DOM are consistent projections.

No projection can retain all the distances. Therefore, we introduce the notion of linear alteration to measure the distortion of distances caused by the different projections.

Linear Alteration = (projected distance – ellipsoid distance) / ellipsoid distance
Linear alteration is expressed in centimetres per kilometre.
For example, the 4 Lambert area projections specific to the NTF were calculated so that the linear alteration is better than 1 per 1000, ie less than 1 meter per kilometre. For the Lambert 93, the linear alteration is from -1 m / km to +3 m / km
The linear alteration is local and variable in each point of the map.

Is linear alteration a source of uncertainty for our data in the information system? This is the case when we do not use high-performance GIS tools. The main softwares used (ArcGis, QGis, …) allow to choose how to measure distances: either on the projected plane or on the ellipsoid. Just choose the second option (ellipsoid) to remove the linear alteration of our measurements. On the other hand, there is almost no mention of the problem caused by the difference between the calculated distances on the ellipsoid and the actual distances, perhaps because there is no technical solution in the GIS …

Conversion between coordinate systems

As we have already mentioned, the conversion between different coordinate systems is provided by the current GIS software, without the user having to deal with it. Just make sure that the definition of the SRC matches that used for the creation of the data, the projection ‘on the fly’ substitutes the user.

Even if this explanation is very hands-on, it has the disadvantage of letting the user believe that any system transformation is neutral and that he does not have to worry about it.

To prove that this is not the case at all, let’s take an actual example. We want to create a Category A network.

Definition of Class A: a structure or section of a structure is classified in class A if the location indicated by its operator deviates from the actual location by at the most 40 cm if it is rigid, or at the most 50 cm if it is flexible (or at most 80 cm in the case of underground civil engineering works associated with a rail or guided transport, built before 01/01/2011);

We equip ourselves with a GPS allowing an accuracy of 10 cm and off we go.

We obtain a GPS position, in Latitude / Longitude, which corresponds to the geodesic system WGS84. Suppose for the sake of our example that this position is:

latitude: 45.1123456789 ° N

longitude: 1,123456789 ° E

Depending on the projected coordinate system we choose to work with, the consequences on our accuracy will not be the same. As we have seen above, old Terrestrial systems had metric accuracy, while the new global systems are between centimetric and decimetric accuracy.

Keep in mind that in a process line, accuracy is the least accurate step.

For now, our GPS data has an accuracy of 10cm.

If I configure my information system in Lambert 93 RGF93 (which, by the way, is an obligation for French public service data since the 2000 decree), I use an accuracy system that is at least equivalent. My network plans will therefore be in the class A standard that I set for myself.

If, however, by habit, we continue to work in Lambert 3 NTF and we prefer to keep our old habits (and process line), I will integrate this data into my information system in Lambert 3, and if I need to communicate my data to a third party, I will revert them to Lambert 93.

Here is the numerical comparison of the two scenarios:

Let’s first look at the bottom path: the GPS positions are transformed into Lambert 93 plane coordinates, and then we re-transform them to geographic coordinates. Both systems have the same centre of the Earth, and the transformation method has decimetric precision. You will observe that the latitudes and longitudes of departure and arrival are identical. The conversion to Lambert 93 did not affect the data acquired in the field.

Now, let’s look at the top way: the GPS positions are transformed into Lambert 3 plane coordinates. The first ones are in WGS84, the second ones in NTF: the Earth centre used by each system are not the same. The transformation method has a metric precision. If we now re-transform the Lambert 3 coordinates into geographic coordinates, the result differs from the input data.

Let’s take the latitude. A degree of latitude = 60 nautical miles = 111 120 meters.

The difference observed between the latitude of departure and arrival is 45.123456789 – 45.123491593 = -0.000034804 °

This represents a distance of -0.000034804 * 111120 = -3.87 meters.

Since we have made two transformations of geodetic system (centre of the Earth), the difference between our Lambert 3 network and the network measured by the GPS would be of the order of 1.9 meters.

Conclusion: we are far from the class A desired.

Conclusion

When we have accuracy constraints in our geographic information system it is essential to analyse and adapt the process line.

Everything we have discussed in this article is exclusively related to the accuracy of the different stages. In terms of accuracy, accuracy or fidelity, this is related to the equipment and measurement method used. What we have discussed applies to an accurate, faithful and accurate measure, or considering ideal conditions of measurement. If this is not the case, the result will be degraded.

As a general rule, coordinate system transformations involving a geodetic system change must be avoided.

And most importantly, even if it sounds obvious to you (I’ve heard it so many times before), remember that data that has a precision of one meter (NTF) does not magically become accurate to just 10 cm by performing a coordinate system change. The Lambert area 1 to 4 NTF data, transformed to Lambert 93 will always have a precision of one meter. Only in the case of our example, where the data is acquired with decimetric precision when transforming Lambert 93 will have a decametric precision.

Once a data is created, its accuracy can only decrease with the treatments it undergoes. Never, ever, its precision can be improved.

For those interested in the subject, here is an interesting reference:

Ashkenazi, V. (1986). Coordinate Systems: How to Get Your Position Very Precise and Completely Wrong. Journal of Navigation,39 (2), 269-278.